2. An Introduction
• Euclid ,sometimes called Euclid of Alexandria , was
a Greek Mathematician, often referred to as the
"Father of Geometry". He was active
in Alexandria during the reign of Ptolemy I (323–283
BC). His Elements is one of the most influential
works in the history of Mathematician, serving as the
main textbook for
teaching Mathematics (especially Geometry) from the
time of its publication until the late 19th or early 20th
century.
3. Euclid’s Elements
• In the Elements, Euclid deduced the principles
of what is now called Euclidean Geometry from
the small set of axioms. Euclid wrote works on:-
• Number Theory
• Perspectives
• Conic Sections
• Spherical Geometry etc.
4. Number theory
• Euclid although is best known for its geometric results,
the Elements also includes number theory. It considers the
connection between perfect numbers and mersenne primes( Euclid-
Euler theorem), the infinitude prime numbers, Euclid’s lemma on
factorization (which leads to the fundamental theorem of
arithmetic on uniqueness of prime factorization), and the Euclidean
algorithm for finding the greatest common divisor of two numbers.
5. Geometry
• The geometrical system described in
the Elements was long known simply
as Geometry, and was considered to be the only
geometry possible. Today, however, that system is
often referred to as Euclidean Geometry .
6. Geometry Works
• The concepts include : Dimension, Angles, Curve,
Diagonals, Parallel, Perpendicular, Vertex,
Symmetry, Similarity, Congruence etc.
• His works are also done on Zero/One-Dimensional,
Two-Dimensional, Triangles and its Properties,
Quadrilateral, Parallelogram, Three-Dimensional,
Circles, Four Dimensional (Tesseract) and Volume,
Perimeter and Area.
7. Euclid’s Perspectives
• Optics is the earliest surviving Greek treatise on
perspective. In its definitions Euclid follows the
Platonic tradition that vision is caused by discrete
rays which emanate from the eye.
• Euclid relates the apparent size of an object to its
distance from the eye and investigates the apparent
shapes of cylinders and cones when viewed from
different angles.
• He proved that for any two unequal magnitudes,
there is a point from which the two appear equal.
8. Perspectives - Definitions
• Things seen under a greater angle appear greater, and
those under a lesser angle less, while those under
equal angles appear equal."
• A point is that which has no part.
• A line has a breathless length.
• The extremities of lines are points.
• A straight line lies equally with respect to the points on
itself.
• One can draw a straight line from any point to any
point.
• Things equal to the same thing are also equal to one
another.
9. Perspectives - Definitions
• A surface is that which has length and breadth only.
• The edges of a surface are lines.
• A plane surface is a surface which lies evenly with the
straight lines on itself.
• When the lines containing angle are straight, the angle
is called rectilinear.
• And when the lines containing angle are straight, the
angle is called rectilinear.
• When a straight line standing on a straight line makes
the adjacent angles equal to one another, each of the
equal angles is right, and the straight line standing on
the other is called a perpendicular to that on which it
stands.
10. Perspectives - Definitions
• An obtuse angle is an angle greater than a right angle.
• An acute angle is an angle less than a right angle.
• A boundary is that which is an extremity of anything.
• A figure is that which is contained by any boundary or
boundaries.
• A diameter of the circle is any straight line drawn
through the center and terminated in both directions by
the circumference of the circle, and such a straight line
also bisects the circle.
11. • A semicircle is the figure contained by the diameter and
the circumference cut off by it. And the center of the
semicircle is the same as that of the circle.
• Rectilinear figures are those which are contained by
straight lines, trilateral figures being those contained by
three, quadrilateral those contained by four,
and multilateral those contained by more than four
straight lines.
• Of trilateral figures, an equilateral triangle is that
which has its three sides equal, an isosceles
triangle that which has two of its sides alone equal, and
a scalene triangle that which has its three sides
unequal.
Perspectives - Definitions
12. • Of quadrilateral figures, a square is that which is
both equilateral and right-angled; an oblong that
which is right-angled but not equilateral;
a rhombus that which is equilateral but not right-
angled; and a rhomboid that which has its opposite
sides and angles equal to one another but is neither
equilateral nor right-angled. And let quadrilaterals
other than these be called trapezoid.
• Parallel straight lines are straight lines which, being
in the same plane and being produced indefinitely in
both directions, do not meet one another in either
direction.
Perspectives - Definitions
13. Perspectives-Notions
• Things which equal the same thing also
equal one another.
• If equals are added to equals, then the
wholes are equal.
• If equals are subtracted from equals, then
the remainders are equal.
• Things which coincide with one another
equal one another.
• The whole is greater than the part.
